To solve the problems on this worksheet, we need to find the solutions (roots) for each quadratic equation by looking at where the graph crosses the x-axis. The x-intercepts of the parabola $y = ax^2 + bx + c$ correspond to the solutions of the equation $ax^2 + bx + c = 0$.
Here is the step-by-step solution for each problem visible in the image:
1. Equation: $x^2 - 2x - 3 = 0$
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Graph Analysis: Look at the first graph. The parabola crosses the horizontal x-axis at two points.
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Identify Points: The curve intersects the x-axis at $x = -1$ and $x = 3$.
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Verification: $(-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0$. And $(3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0$.
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Solutions: $x = -1, 3$
2. Equation: $x^2 + 4x + 3 = 0$
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Graph Analysis: Look at the second graph.
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Identify Points: The curve intersects the x-axis at $x = -3$ and $x = -1$.
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Verification: $(-3)^2 + 4(-3) + 3 = 9 - 12 + 3 = 0$. And $(-1)^2 + 4(-1) + 3 = 1 - 4 + 3 = 0$.
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Solutions: $x = -3, -1$
3. Equation: $-x^2 + 2x + 8 = 0$
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Graph Analysis: Look at the third graph (top right). This parabola opens downward.
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Identify Points: The curve intersects the x-axis at $x = -2$ and $x = 4$.
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Verification: $-(-2)^2 + 2(-2) + 8 = -4 - 4 + 8 = 0$. And $-(4)^2 + 2(4) + 8 = -16 + 8 + 8 = 0$.
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Solutions: $x = -2, 4$
4. Equation: $x^2 - 6x + 9 = 0$
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Graph Analysis: Look at the fourth graph (second row, left).
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Identify Points: The vertex of the parabola touches the x-axis at exactly one point: $x = 3$.
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Verification: $(3)^2 - 6(3) + 9 = 9 - 18 + 9 = 0$.
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Solutions: $x = 3$
5. Equation: $x^2 - 4x - 5 = 0$
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Graph Analysis: Look at the fifth graph (second row, middle).
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Identify Points: The curve intersects the x-axis at $x = -1$ and $x = 5$.
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Verification: $(-1)^2 - 4(-1) - 5 = 1 + 4 - 5 = 0$. And $(5)^2 - 4(5) - 5 = 25 - 20 - 5 = 0$.
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Solutions: $x = -1, 5$
6. Equation: $2x^2 + 7x + 3 = 0$
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Graph Analysis: Look at the sixth graph (second row, right).
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Identify Points: The curve intersects the x-axis at $x = -3$ and $x = -0.5$ (or $-\frac{1}{2}$).
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Verification: $2(-3)^2 + 7(-3) + 3 = 18 - 21 + 3 = 0$. And $2(-0.5)^2 + 7(-0.5) + 3 = 2(0.25) - 3.5 + 3 = 0.5 - 3.5 + 3 = 0$.
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Solutions: $x = -3, -\frac{1}{2}$
7. Equation: $x^2 - 2x - 15 = 0$
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Graph Analysis: Look at the seventh graph (third row, left).
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Identify Points: The curve intersects the x-axis at $x = -3$ and $x = 5$.
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Verification: $(-3)^2 - 2(-3) - 15 = 9 + 6 - 15 = 0$. And $(5)^2 - 2(5) - 15 = 25 - 10 - 15 = 0$.
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Solutions: $x = -3, 5$
8. Equation: $x^2 + 2x - 8 = 0$
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Graph Analysis: Look at the eighth graph (third row, middle).
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Identify Points: The curve intersects the x-axis at $x = -4$ and $x = 2$.
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Verification: $(-4)^2 + 2(-4) - 8 = 16 - 8 - 8 = 0$. And $(2)^2 + 2(2) - 8 = 4 + 4 - 8 = 0$.
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Solutions: $x = -4, 2$
Final Answer:
1. $x = -1, 3$
2. $x = -3, -1$
3. $x = -2, 4$
4. $x = 3$
5. $x = -1, 5$
6. $x = -3, -\frac{1}{2}$
7. $x = -3, 5$
8. $x = -4, 2$
Parent Tip: Review the logic above to help your child master the concept of solving quadratic equations by graphing worksheet answers.
